V.V. Samarin
Joint Institute for Nuclear Research, Dubna samarin@jinr.ru
The aim of report is application of based quantum mechanics equations for neutron transfers description.

Motivation
•
Neutron transfers in the low energy nuclear reactions allow us to obtain new isotopes of atomic nuclei with increased neutron content.
• The probability of neutron transfer is highest during so-called grazing nuclear collisions. In this case the distances between the surfaces of the atomic nuclei do not exceed the range of the action of nuclear forces (1 –2 fm).
• The most probable transition is the one between the nuclei of the external, most weakly bound neutrons.
• A new possibility for theoretical study of this reactions is provided by numerical solution for the non-stationary Schrodinger equation for external neutrons [1].
• In this study the spin-orbital interaction and Pauli's exclusion principle were taken into consideration for spherical and deformed nuclei.
1. V.Samarin, V. Zagrebaev, Walter Greiner. Phys. Rev., C 75, 035809 (2007) .
2

6
197
2.
6 Не + 208 Pb
18
48
48
238
More detail and less detail
2. Spherical and deformed nuclei shell model for external nucleons
3

Time-
Timedependent Schrödinger equation with spin-orbital interaction dependent Schrödinger equation with spin-orbital interaction V
ˆ
LS
 
( )
2 b i
 t




1





2
2 m
 
( , )

V
ˆ
LS
( , )







1
2

 b

R
2
0

2
2
0

0, 022 R
0
2 
R
0
=1 fm in Cartesian coordinates i

 t
1

2
    
2 m
 i b
 
V
2

 y
 z
2




V

 z
 y
2 i b
2
 
V x
 y
1



 b
2




V x

 z
2



V y


V
 x
1




 z
 x
2

, i

 t
  
2

2
2 m
 


 
2 i b
2
 
V x
 y
2


V y

    x
2



 i b
2
 

V y

 z
1


V z
 y
1


 b
2



V x

 z
1


V z
 x
1

.
is numerically solved by difference method [1-3] for external neutrons of spherical nuclei at their grazing collisions with energies near to a Coulomb barrier.
1. Samarin V. V., Samarin K. V. // Bull. Russ. Acad. Sci. Phys. 2010. V. 74. P. 567.
2. Samarin V. V., Samarin K. V. // Bull. Russ. Acad. Sci. Phys. 2011. V. 75. P. 964.
3. Samarin V. V., Samarin K. V. // Bull. Russ. Acad. Sci. Phys. 2012. V. 76. P. 450 .
4

Deformed nuclei shell model
Schrödinger equation for the nucleon energy levels and wave functions at arbitrary axial-symmetrical field with spin-orbit interactionare calculated by a numerical solution of a Schrödinger equation for an arbitrary axial-symmetrical field with spin-orbit interactions, basedon decomposing on Bessel functions and difference scheme along internuclear axis.
Samarin V.V. Phys. Atom. Nucl., 2010, 73 , p. 1416.




2
    
2 m
( , ) b

1
V z i V
2
2
    
2 m
( , ) b

1
V z i V
2






 
1
 
2 i b
 e

2 b
2
 i





 i V


 i e i V


 
1 f

1
  i

1 2)
    f

2

 z
 z z



V z

V z







1
V z

 



1
V z

 






   
2 1
   
1

2 i

1 2)
  f

1
  k
N 

0 p k
 z y k


   
1 2
 
0,1, f

2
  k
N 

0 q k

1 z y k

1
 

0
0 y k

 y n

( ) d kn
5

6 Не + 208 Pb, Е cm
Timedependent Schrödinger equation solution is the way of visual simulation of neutron transfer for reactions with halo nuclei 6 Не.
At first, for frontal and grazing collisions with energy below
Coulomb barrier you can see, that
6 Не neutron wave function of 1p
3/2 state is arranged between projectile and target nuclei. There is stable space structure, like to 3d state of
Au or Pb, with zero angular momentums projection to internuclear axis.
6 Не + 197 Au, Е cm
6

6 Не + 208 Pb, Е cm
At second, for frontal and grazing collisions with energy in vicinity of
Coulomb barrier you can see, that 6 Не neutron wave function of 1p
3/2 state is arranged between projectile and target nuclei with some space structure too.
6 Не + 197 Au, Е cm
7

At next, for grazing collision with energy above of Coulomb barrier you can see, that 6 Не neutron wave function of 1p
3/2 state is arranged between projectile and target nuclei with stable space structure like to rotated 3d state of Au.
6 Не + 197 Au,
Е cm
8

At last, for grazing collision with energy more above of
Coulomb barrier you can see, that 6 Не neutron wave function of 1p
3/2 state is arranged between projectile and target nuclei with some space structure like to rotated 3d state of
Au too.
6 Не + 197 Au,
Е cm
9

If distances between nuclear centers and surfaces is decreasing, potential barrier between two potential walls is decreasing too ( Fig. 1), transfer probability is increasing ( Fig. 2). This reason lead to cross section of neutron transfer at 6 He+ 197 Au, which satisfactorily agrees with experimental data (Fig. 3).
Spacing interval between nuclear surfaces
Fig. 1.
p
Fig. 2.
Probabilities p of the transfer of the external neutron of the 6 Не as functions of the minimum distance s between the surfaces of 6 Не, 197 Au nuclei for the energies in the center of mass system near barrier energies from 18 to 22 MeV
( ), 30 MeV ( ) and 60 MeV ( ).
Fig. 3.
10

We may illustrate semiclassically dominant neutron transfer for neutron orbits with zero angular momentums projection to internuclear axis, which touching each other.
Neutron energy and momentum for classical orbit in projectile equal approximately to neutron energy and momentum for classical orbit in target. In this case angular momentums relatively to centers of nuclei will be proportional to radii of nuclei (Fig. 1).
L
2
 pR
2
 pR
1
R
2

L
1
R
2
R R
1 : 1,
1

2 : 3
1

L d
1 2
For reaction 6 Не + 197 Au, in frontal collision dominant transfer is: 1p
3/2
( 6 Не) 
3d
5/2
,
3/2
(
Calculated by timedependent Schrödinger
197 Au).

L
1
1p
3/2
6 Не
R
1
R
2
197 Au equation occupied states probabilities in Au for stripping neutron with full momentum projection
 Fig. 1 p
3d
5/2,3/2 are shown on Fig. 2.
p 0,2 0,2 E=18 MeV
 =1/2 and  =3/2
 =1/2
3d
3/2
E=14 MeV p
 =1/2 and  =3/2
 =1/2 3d
5/2
2g
7/2   
2 MeV
0,1 0,1
4s
1/2
3p
Fermi level
3/2
1i
13/2
3p
1/2
2g
9/2
3d
5/2
1i
11/2
2g
7/2
4s
1/2
3d
3/2
0,0
20 22 24 26 28
Level number in
197
Au
0,0
Fermi level
3p
3/2
1i
13/2
3p
1/2
2g
9/2
20
Fig. 2
22
1i
11/2
24 26
11
Level number in
197
Au
28

Pauli's exclusion principle limit transfers to occupied states for reactions 18 O+ 48 Ca.
Preliminary: unlimited transfers
Begin of collision
18 O
48 Ca
End of collision
Neutron states in spherical shell model
12

Pauli's exclusion principle limit transfers to occupied states for reaction: 48 Ca+ 248 U
Preliminary: unlimited transfers
Begin of transfer
48 Ca
238 U
2g
9/2
48 Ca
238 U
End of collision
Neutron states in spherical shell model
Neutron states in deformed shell model are studied in next slides
Begin of transfer
1f
7/2
End of collision
2g
9/2
Begin of collision
End of collision
48 Ca
3d
5/2
238 U
13
48 Ca 238 U

Pauli's exclusion principle were taken into consideration by:
1 . exception with transfer to occupied states in “frozen” nuclei shell structures
(simple approximation).
2. time dependent many body wave function (M=2, 3)

M
(more correct approximation).
 r
1
, n
,
 
1
M !
det





1
M
 
 

1


N

M

N





P
M
 r
1
, r
M
, t
 
M
 r
1
, r
M
,

1
...
dV
M
Fig. 1. Probabilities of neutrons pick-up (solid curves) and stripping
(dashed curves) for 48 Са at reactions 48 Са + 18 O (curves 1) и
48 Са + 238 U (curves 2) as function on minimum value of internuclear distance
At reaction 48 Ca+ 238 U probabilities of neutrons stripping and pick-up are commensurable .
Fig. 2. Probabilities of neutrons pick-up (solid curves) and stripping (dashed curves) for 48 Са at reactions 48 Са + 238 U as function on minimum value of internuclear distance 14

Visual simulation of neutron transfers at reaction 18 O+ 48 Ca.
Two external neutrons:
1d
5/2
1f
7/2 from 18 O and from 48 Ca with moment projection

=1/2, 3/2 are took into account
Visual simulation of neutron transfers at reaction 48 Ca+ 238 U
Three external neutrons:
1f
7/2 from
2g
9/2
, 1i
11/2
48 Ca and from 238 U with moment projection

=1/2, 3/2 are took into account
15

Nucleons transfers in low-energy nuclear reactions
40,48 Ca+ 238 U with deformed nucleus 238 U a b a ) Some upper energy levels for neutron states with module
 of total angular momentum projection to symmetry axis at hypothetic nuclei 238 U with quadrupole deformation b
2 b
2
= 0.215, b
4 and octupole deformation b
4
= b
2
/2 (a) and at real nucleus 238 U with
= 0.095 ( b ), dashed lines correspond unoccupied levels
16

Neutron pick-up at reaction 48 Ca+ 238 U
Visual simulation of neutron pick-up at reaction 40 Ca+ 238 U during frontal collision at energy in the center of mass system E =192 MeV.
In the beginning external neutron of 238 U is in initial state 1j
15/2 with angular momentum projections on symmetry axis

=5/2.
Angle between symmetry axes of deformed nucleus
238 U and initial velocity of
40 Ca nucleus equal 45 o .
17

40
238
238
a b a ) Probability density of the external neutron of 238 U for initial state 1j
15/2 with angular momentum projections on symmetry axis

=5/2 during frontal collision with the 40 Ca at energy in the center of mass system E =192 MeV. Angle between symmetry axes of deformed nucleus 238 U and initial velocity of 40 Ca nucleus equal 45 o .
b ) The probabilities of neutron pick-up at reaction 40 Са+ 238 U as a function of minimum distance s between nuclear surfaces. Angles between symmetry axes of deformed nucleus
18
238 U and initial velocity of 40 Ca nuclei equal 45 o (solid line) and 90 o (dashed line).

Neutron stripping at reaction 48 Ca+ 238 U
Visual simulation of neutron stripping at reaction 48 Ca+ 238 U during frontal collision at energy in the center of mass system E =175 MeV.
In the beginning external neutrons of 48 Ca is in initial state 1f
7/2
. Angle between symmetry axes of deformed nucleus 238 U and initial velocity of 48 Ca nucleus equal 0 o .
19

Neutron stripping at reaction 48 Ca+ 238 U
Visual simulation of neutron stripping at reaction 48 Ca+ 238 U during frontal collision at energy in the center of mass system E =192 MeV.
In the beginning external neutrons of 48 Ca is in initial state 1f
7/2
. Angle between symmetry axes of deformed nucleus 238 U and initial velocity of 48 Ca nucleus equal 45 o .
20

48
238
238
a b a ) Probability density of the external neutrons of 1f
7/2 shell of 48 Ca during a frontal collision with the 238 U at energy in the center of mass system E =192 MeV. Angle between symmetry axes of deformed nucleus 238 U and initial velocity of 40 Ca nucleus equal 45 o .
b ) The probabilities of neutron stripping at reaction 48 Са+ 238 U (a) as a function of minimum distance s between nuclear surfaces. Angles between symmetry axes of deformed nucleus 238U and initial velocity of Ca nuclei equal 45 o (solid line), 90 o (dashed line) and 0
(dotted line).

Conclusion
•
A new possibility for theoretical study of this reactions is provided by numerical solution for the nonstationary Schrödinger equation for external neutrons.
• In this study the spin-orbital interaction and Pauli's exclusion principle were taken into consideration. Timedependent Schrödinger equation is numerically solved by difference method for external neutrons of spherical nuclei 6 He, 18 O, 48 Са and deformed nucleus 238 U at their grazing collisions with energies near to a Coulomb barrier.
•
The probabilities of transfer of neutrons at reactions 6 He+ 197 Аu, 18 O+48Ca, 40,48 Ca+ 238 U are determined as function on minimum internuclear distances.
• The calculation results of cross section for formation of the 198 Au isotope in the 6 Не+ 197 Au reaction agree satisfactorily with the experimental data in vicinity of the Coulomb barrier.
• At reactions 6 He+ 197 Аu, 18 O+ 48 Ca, neutrons are predominantly transferred from a smaller nucleus to the greater nucleus. At reaction 48 Ca+ 238 U probabilities of neutrons stripping and pick-up are commensurable.
• Nonstationary quantum approach applied in this work may be used for internal nucleons too. It may be useful for nucleons transfer experimental data analysis.
22

23